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What is Cube and Cube Root?

                      

Introduction –  

In geometry, a solid three dimensional object which has equal sides is called cube. It has 6 sides and 12 lines. But here we study about cube in arithmetic.

In number, the term cube is three times multiplication of any number itself. For example take a number 2, cube of 2 means three times multiplication of 2 itself.

                                               2 x 2 x 2 = 8

It is denoted by 2³ as you write in square.  2³ = 8 and said like 2 cube as 2² = 4 is said two square. 

What is Perfect Cube?

A perfect cube is a number that is the cube of an integer like Perfect Square. When you factorize a cube via prime factorization, then each factor should be multiplied thrice itself. If the prime factors are 2 time or four times, then it is not a perfect cube. For example – 81. 125 is a perfect cube since 125 = 5 x 5 x 5 = 5³

How to know the given number is perfect or imperfect cube?

By prime factorization – when factorize any number then the factor should be three time multiplied by itself as we have seen in the above example. Some examples of perfect cubes are :- 



                                             1 = 1 x 1 x 1, 


                                              8 = 2 x 2 x 2, 


                                             27 = 3 x 3 x 3, 


                                             64 = 4 x 4 x 4 x 5, 


                                             125 = 5 x 5 x 5, 


                                              216 = 6 x 6 x 6, 



                                             343 = 7 x 7 x 7, so on.

Imperfect cube – The number is said to be imperfect cube if the prime factor is not multiplied by three times itself. For example, take a number 100, lets factorize it,

                                                     2| 100 |

                                                     2| 50 |

                                                     5| 25 |

                                                     5| 5 |

                                                       | 1 |

               = 2 x 2 x 5 x 5 = 100

This is not perfect cube because both factors are not multiplied three times itself. Hence it is an imperfect cube.

Make Imperfect cube to perfect cube:- You may asked question to make  a imperfect cube into perfect cube by multiplying or dividing. The question could be like it.

Question 1 - What is the least number that make 100 perfect cube by multiplying.

Solution:- As we have seen in the above example - 100 is not perfect cube because its both prime factors 2 and 5 are multiplied only two times.

2 x 2 x 5 x 5, if we added one more 2 and 5 here, it will become perfect cube. It means, we multiply 100 by 10 (2 x 5 ) to make it perfect cube.

2 x 2 x 2 x 5 x 5 x 5 = 2 x 5 = 10

Hence 10 x 10 x 10 = 1000 = 10³ 

Question 2:- Make 392 a perfect cube.

Solution:-  By prime factorization,

                                            2 | 392 |

                                            2 | 196 |

                                            2 | 98 |

                                            7 | 49 |

                                            7 | 7 |

                                               | 1 |

              
                                       = 2 x 2 x 2 x 7 x 7

Here first factor 2 is multiplied thrice while second factor 7 is multiplied twice. If second factor 7 would have been multiplied thrice, then the new number would have been a perfect cube. It means if we multiply number 392 by 7 we get the perfect cube.

2 x 2 x 2 x 7 x 7 x 7 = 2 x 7 = 14; 14³

It means if 392 is multiplied by 7, it will become perfect cube

 = 392 x 7 = 2744 = 14³ = required cube.

Question 3 :- What is least number that divides the given number x to make it perfect cube. For example the given number be 81

Solution – Let us factorize 81

                                                3 | 81 |

                                                3 | 27 |

                                                3 | 9 |

                                                3 | 3 |

                                                   | 1 |

                                                  


                                    = 3 x 3 x 3 x 3

Here the factor is multiplied four times while it should have been multiplied only three times, to be a perfect cube. There is one extra 3. Hence to make it perfect cube, we shall dived the given number 81 by 3.

81 / 3 = 27, the required least number to make perfect cube is 27 = 3³ 

What is Cube Root?

The cube root is just opposite of cubing a number and it is obtained by prime factorization. Since 5³ = 125, the cube root of 125 is 5. The cube root of a perfect cube is an integer.   

Cube root is denoted by mathematical sign ³√.

Finding cube root by prime factorization

Question 1:- Find cube root of 64 or ³√64

Solution:-

                                                4 | 64 |

                                                4 | 16 |

                                                4 | 04 |

                                                   | 01 |

                                         = 4 x 4 x 4 = 4

 Here cube root of 64 is 4.

Question 2:- What is cube root of 216 or ³√216?

Solution:-  

                                                 2 | 216 |

                                                 2 | 108 |

                                                 2 | 54 |

                                                3  | 27 |

                                                 3 | 9 |

                                                 3 | 3 |

                                                    | 1 |   

We can write – 2 x 2 x 2 x 3 x 3 x 3 = 216

Since, 2 and 3 are multiplied three time itself hence the prime factor is 2 and 3.

Therefor, cube root of 216 = 2 x 3 = 6. That means, if you will multiply 3 times (3 x 3 x 3), you’ll get 216.

It is also possible to find the cube root of a negative number. For example, the cube root of – 125 is – 5 or – 216 is – 6.

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more maths

What is Calculus?

What is Calculus?

The Calculus is a branch of mathematics deals with change. The change takes place everywhere in the universe. It happens around you and in your life. Babies grow, Youngman grows, business grows, and your bank balance grows or diminishes.

The number of friends on your social media (Facebook & Twitter) gets increase or decrease as well as number of like & share increase or decrease.

Film stars’ popularity increases or decreases as per their latest movie’s performance. Virat Kohli or Rohit Sharma’s follower increases on twitter as he plays another sterling inning. Your position keeps on changing. Amount of internet data of your mobile phone/laptop keeps on decrease according to your uses.

In the above mentioned activities everything keeps on change; somewhere it increases and somewhere decreases but one thing is certain that things get change. The calculus is the branch of mathematics that enables us to find out each minute change.

Calculus begins when Arithmetic fails:-  Suppose you drive a motor car with an average velocity of 60 km/h. Suddenly you decreases car’s speed to 0 km/h or apply break but the car does not stop at that place. It stops after a few minutes and after travelling some distance.

According to Arithmetic, the car should stop at that place where the break applied or engine snoozed but practically, the car keeps on moving and stop after travelling some distance.

Here, Arithmetic fails to explain the consequence but Calculus tells us when and where the car will stop. How much distance it will travel and how much time it will take to come into rest.

In short, Calculus tells us about all change whether it is smaller or bigger. There is mainly two ideas of Calculus – Derivative (Differential) and Integral.

Derivative:- It tells us how fast something is changing. It was discovered in 1600s in England and Germany. It is also called differential (calculus).

Integral:- How much something is accumulating. It was discovered in 250 BC in Greece.

This is two core ideas of Calculus and rest of thing is details of it. We read about how to get derivatives and how to integrate but core ideas behind it are these two ideas.

Pleasant twist in History:- Both concept were born in separate time and places. But they proved to be relatives. This relationship revolutionized Physics and mathematics forever.    

Fastest way to find out Square up to 999 in a second?

Fastest way to find out square up to 999: Get square of 1 to three digits in a second. 

To find out square of any two digits or above number by convention method is not an easy task. Usually, it takes lots of time and tedious multi level multiplication as a result students get tire and bore.

They need to get some short-cut method to find out square of big numbers as they move to upper classes. There are some examples of difficult square that can be done verbally and needs pen and paper like; 37² 69², 357², 697² 949² etc.

Don’t worry, Today, mathematics made easy brings some useful method that will make your calculation quite easy. You can find out square of difficult numbers of two or three digits number after little practice.

For Two (2) Digit square

Method 1:-

Step one:- Suppose you wanted to find out square of 32 

 Write the desired number with square sign. It is applicable in 2 digit numbers square.

(32)² 

Step 2:- Multiply each digit and exponent power in this way

3 x 2 x 2 =12

Now, write square of first two numbers from your left side and add the number (12) right hand side of equals sing, leaving one’s digit place empty in this way 

3 x 2 x 2 = 12

                        09 04

and add them     12

1024 this is your desired number or answer. Do you doubtful about result? Lets us check it by long method.

  32

x 32 .

    64

  9 6..

1024

This is your desired result which is equal to result found via shortcut method. Is it not effect and time consuming in examination, "Isn't It"?. 

Method 2:- This is universal method and applicable up to 999. It will also help you to find the square up to 3 digits in a second without pen and paper.

What is method?

Step 1:- To get the desire square, make it nearest

Suppose you wanted to get the square of 87. Make it closed to that number that has 0 at one’s place. The nearest number is 90. You need to add 3 to 87 to make it 90.

Now, subtract 3 from 87, the number comes out 84. Now write it in this way,

90 x 84

Now, write square of 3 that is 9; in this way along with previous number

90 x 84 | 09

Divide 90 by 10 to remove 0 from one’s place and cancel it.

90/10 x 84 | 09

Now, multiply 9 with second number 84 and place 9 from 09. Add the carried digit at double digit place 0 with one digit place number of product of 9 x 84 that is 756.

 9 x 84 | 09 = 756 | 09

Now, 9 will stay at one’s digit place and second digits place number 0 will carry and added to 756.

7569 and this is desired square of 87.

Isn’t interesting? Do more practice to make faster.  The more you practice the more you get quickest.

Find out Square of 3 digits number

Square of 213 = ?

Subtract 13 and add 13 and write square of 13 at extreme right side.

200 x 226 | (13)² = 169

200/100 x 226 | 169

2 x 226 | 169

452 | 169

 = 45369

ExceptionalCase:- Find square of 747 

Sol:- Find nearest number having 0 at one’s place i.e., 750. Now subtract 3 from it and add 3 to it.

750 X 744 | 09

Now, cancel 0 from 750 by dividing it by 100 and multiply with 744

750/100 X 744 | 09

75/10 X 744 | 09

15/2 X 744 |09

We can write 15/2 as 30/4 for convenience to do the calculation. It won’t make any impact on this process but make calculation eaiser.

 30/4 X 744 | 09

30 X (divide 744 by 4) 186 | 09

30 X 186 | 09

= 558009

Question: Find square of 888, 257, 668, 572, 345, 267,879 etc.

Electric Current | Hindi

What is the Infinite sum of Series 1/1 + 1/3 + 1/6 + 1/10 + 1/15 + ?

Linear equaation, Irrational Number, Property of Rational Number

Question:-What is the infinite sum of the series 1/1 + 1/3 + 1/6 + 1/10 + 1/15 + ?

Answer:-  Let Tn represents the n-th term in this series.

T1 = 1/(1)

T2 =,1/(1+2)

T3 = 1/(1+2+3)

T4 = 1/(1+2+3+4)

T5 = 1/(1+2–3+4+5) and so on

-> Tn = 1/(1+2+3+……..+n)

= 1/ [ n(n+1)/2 ] = 2/n(n+1)

Tn = 2 / [ n(n+1) ]

= 2 * [ (1/n) - 1/(n+1) ]

If you add all the terms you will get >

2 * [ 1/1 -1/2 + 1/2 -1/3 + 1/3 - 1/4 +1/4 - 1/5 + 1/5 - 1/6 +…….]

= 2  Answer

Solve by Raj

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What is Trigonometrical function?

 

Trigonometrical  function:In trigonometry there is importance of all thing whether it is line, angle, identities or function. Trigonometrical function has some perticular importance. If you are familiar with these function and formulas; you can easily solve problem. 

So, here, we gathered all trigonometrical function and formulas. This has been done for the easiness of students or math lover. Hope, you'll like it and suggest others to at least read once.

 

 

Reciprocal identities

 

Sin x = 1/cosec x        cos x = 1/sec x           tαn x = 1/ cot x

cosec x = 1/sin x        sec = 1/cos x               cot x = 1/tαn x

Pythagorean Identities

sin² x + cos² x = 1

1 + tαn² x = sec² x

  1 + cot² x = cosec² x

Quotient Identities

tαn x = sin x/ cos x,

cot x = cos x/ sin x

Co-Function Identities

sin (π/2 – x) = cos x

cos (π/2 – x) = sin x

tαn (π/2 – x) = cot x

cosec (π/2 – x) = sec  x

sec(π/2 – x) = cosec x

cot (π/2 – x) = tαn x

Even – Odd Identities

Sin (- α) = - sin α cos(- α) = cos α, tαn( - α ) = - tαn α

Cosec ( - α) = - cosec α, sec (- α) = sec α cot (- α) = - cot α

Sum – Difference formulas

sin (x + y) = sin x cosy + cos x sin y

Cos (x + y) = cos x cox y -+ sin x sin y

tan (x + y ) = tan x + tαn y/ 1-+ tan x. tan y

Double Αngle Formulas

sin (2x) = 2sin x cosx

cos (2x) = cos2 x – sin 2 x

 

              = 2cos2 x – 1 = 1 2 sin2 x

tan (2x) = 2tan x/ 1 – tan2 x/1 tan2 x

Power-Reducing / Half angle formulas

Sin2 x = 1 – cos (2x)/2

Cos2 x = 1 + cos (2x)/2

tan2 x = 1 – cos (2x)/1 + cos (2x)

Sum to Product Formulas

Sin x + sin y = 2 sin (x + y/2) cos (x – y/2)

Sin x – sin y = 2 cos (x + y/2) sin (x – y/2)

cos x + cos y = 2 cos (x + y/2) cos (x – y/2)

cos x – cos y =  – 2sin (x + y/2) sin (x – y/2)

Product to Sum Formulas

sin x.sin y = ½ [cos (x – y) – cos (x + y )]

cos x cos y = ½ [cos (x – y) + cos (x + y)]

sin x cos y = ½ [sin (x +y) + sin (x – y)]

cos x cos y = ½ [sin (x – y) – sin (x – y)]

How to find square by simple method?

What is square, perfect square & its property? 

 What is Decimal fraction & How can get it

How Important is Number System in Math & Life

Number System: Facts and Useful Formula

Puzzle Question – If  Two third of a Number….what is Number?

What is Euclid’s Division or Lemma, Real Number, Part-1

Rationalization of Surd or Radical-Part3

Simplest form of Radical or Surd-Part II

What is Radical or Surd?

Prove 3√6 is a rational number

Prove√7 is not a rational number

Prove √3 is not a rational Number

Limit-Part II

What is Limit in Calculus? Introduction

Factorization of Polynomial Part V

Factorization by Splitting middle term Part IV

Factorization Using Identities – Part III

Factorization Using Identities – Part II

Factorization made Easy- Part I

Algebraic Expression Part – 2

Algebraic Expression

Linear Equation Part - 4

Linear Equation Part – 3

Linear Equation Part – 2

Linear Equation in one variable – Part -1

Real Numbers

Irrational Number

Rational Number – 4

Properties of Rational number – 3

Properties of Rational Number – 2

Properties of Rational Number

Mathematics Importance of Rational Number

Importance of Rational Number